Uncertainty Relation Based on Skew Information with Quantum Memory

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by  Zhihao Ma

SCIENCE CHINA
Physics, Mechanics & Astronomy
1
Vol. No.:
doi:
UncertaintyRelationBasedonSkewInformationwithQuantumMemory
ZhihaoMa1,ZhihuaChen2,andShao-MingFei∗3,4
1SchoolofMathematicalSciences,ShanghaiJiaotongUniversity,Shanghai,200240,China
2DepartmentofAppliedMathematics,ZhejiangUniversityofTechnology,Hangzhou,310014,China
3SchoolofMathematicalSciences,CapitalNormalUniversity,Beijing100048,China
7 4Max-Planck-InstituteforMathematicsintheSciences,04103Leipzig,Germany
1
0 PACSnumber(s): 03.67.-a,75.10.Pq,03.67.Mn
2

n DearEditors, tum uncertainty of H in ρ. Luo introduced another quantity
√
Ja Uncertainty principle is one of the most fascinating features in [7], Jρ(H) = 12Tr[({ ρ,H0})2], where {X,Y} is the anti-
of the quantum world. It asserts a fundamental limit on the commutator, H0 = H −Tr(ρH)I with I theidentityoperator.
9
precisionwithwhichcertainpairsofphysicalpropertiesofa Thefollowinginequalityholds[7],

h] pnaerotuicslley,ksuncohwans. pTohseitiuonncaenrtdaimntoympernintucimpl,ecahnasnaotttrbaectseimduclotna-- (cid:113)I(ρ,R)J (R)(cid:113)I(ρ,S)J (S)(cid:62) 1|Tr(ρ[R,S])|2. (1)
p ρ ρ 4
siderable attention since the innovation of quantum mechan-
-
nt icsandhasbeeninvestigatedintermsofvarioustypesofun- (cid:112)I(ρ,R)Jρ(R) can be regarded as a kind of measure for
a certainty inequalities: as informational recourses in entropic quantum uncertainty. Hence we define UN(ρ,R) :=
u terms,bymeansofmajorizationtechniqueandbasedonsum (cid:112)I(ρ,R)J (R). Then we define the uncertainty of ρ asso-
q ρ
ofvariancesandstandarddeviations. ciated to the projective measurement {φ } as: UN(ρ) =
1 [ For a pair of observables R and S, the well-known (cid:80)kUN(ρ,φk) = (cid:80)k (cid:112)I(ρ,φk)Jρ(φk), wkhere φk = |φ{kφ(cid:105)k(cid:104)}φk|
Heisenberg-Robertson uncertainty relation [1] says that, andψ = |ψ (cid:105)(cid:104)ψ |aretherankonespectralprojectorsoftwo
v k k k
6 Vρ(R).Vρ(S) (cid:62) 14|Trρ[R,S]|2,where[R,S] = RS −SRisthe non-degenerate observables R and S with eigenvectors |φk(cid:105)
2 commutatorandVρ(R)isthestandarddeviationofR. Theen- and|ψk(cid:105),respectively.
2 tropiesserveasappropriatemeasuresoftheinformationcon- Now we consider the case of bipartite state ρ in tensor
AB
2 tent. It is also used to quantify the quantum uncertainty: the space H ⊗H [8]. Recallthatquantumdiscordisakindof
0 A B
sumoftheShannonentropyoftheprobabilitydistributionof quantum correlation that is different from the entanglement
.
01 theoutcomesisnolessthanlog2 1c [2]whenRandS aremea- and has found many novel applications [9]. A bipartite state
sured.Theterm1/cquantifiesthecomplementarityofthetwo ρ is of zero discord if and only if it is a classical-quantum
7 AB
1 observablesRandS. Ithasbeenprovedthattheentropyun- correlatedstate(CQstate). Besidesthedefinitionoftheorig-
: certaintyrelationsdoimplytheHeisenberg’suncertaintyrela- inaldiscord,therearesomeotherdiscord-likemeasuresshar-
v
i tion. ing the sameproperties such that their valuesare zero iff the
X
Concerningbipartitesystems,theauthorsin[3]provideda stateisaCQstate.Inthisletterwedefineanotherdiscord-like
r bound on the uncertainties which depends on the amount of measure.LetObeanyorthogonalbasisspaceinHilbertspace
a
entanglementbetweenthemeasuredparticle Aandthequan- H . Let |φ (cid:105) be an orthogonal basis of H and φ = |φ (cid:105)(cid:104)φ |
A k A k k k
tummemoryB. Theresultof[3]wasfurtherimprovedtode- theorthogonalprojectionsonH .Wedefinethequantumcor-
A
pendonthequantumdiscordbetweenparticlesAandBin[4]. relationofρ as:
AB
Recentlytheauthorsin[5]obtainedentropicuncertaintyrela-
(cid:16) (cid:17) (cid:88)
tionsformultiplemeasurementswithquantummemory. Q ρAB =min [I(ρAB,φk⊗IB)−I(ρA,φk)], (2)
O
Thequantumuncertaintyrelationcanbealsodescribedin k
termsofskewinformation. I(ρ,H)=−1Tr(cid:104)√ρ,H(cid:105)2isintro- wheretheminimumistakenoveralltheorthogonalbasesin
2
ducedtoquantifythedegreeofnon-commutativityofastateρ H ,ρAisthereducedstateofsystemA.
A
andanobservableH,whichisreducedtothevarianceV (H) Fromtheinequalityin[10]thatI(ρAB,X⊗I)(cid:62) I(ρ ,X)for
ρ A
when ρ is a pure state [6]. It can be interpreted as quan- anybipartitestateρAB andanyobservable X on HA,wehave
1*Correspondingauthor(email:[email protected])
(cid:13)c ScienceChinaPressandSpringer-VerlagBerlinHeidelberg phys.scichina.com link.springer.com
SciChina-PhysMechAstron () Vol. No. -2
(cid:16) (cid:17) (cid:16) (cid:17)
thepropertythatQ ρAB (cid:62) 0. Moreover, Q ρAB = 0ifand Take σ and σ as the two observables. For our theorem
x z
only if ρ is a CQ state by using the method in proving the we have the values of the left hand side and the right hand
AB (cid:113) √ √ √
theor(cid:16)em1(cid:17),property(1)of[11]. side of (3), 5−2 3−3p2−2p(1−p+ 3−3p2) and 2−p− 3−3p2, re-
Q ρAB has a term of measurement on the subsystem A, 3 3
spectively. While the left hand side of the theorem in [7]
which gives an explicit physical meaning: it is the minimal (cid:112) (cid:112)
difference of incompatibility of the projective measurements is 91(2 − p − 3−3p2)(4 + p + 3−3p2) and the right
hand side is 0. From the result in [4], the left hand side is
onthebipartitestateρAB andonthelocalreducedstateρA. It
−2(2−p)log 2−p−2(1+p)log 1+p−2,theboundisthesameas
quantifiesthequantumcorrelationsbetweenthesubsystemsA 3 2 6 3 2 6
thelefthandside,seeFig. 1forcomparision.
andB.
TheoremLetρABbeaquantumstateonH ⊗H ,{φ }and
A B k
y
{ψ }denotetwosetsofrankoneprojectivemeasurementson
k 2.0
HA. Thenthefollowinguncertaintyrelationholds:
(cid:16) (cid:17) (cid:16) (cid:17)
UN ρAB +UN ρAB
(cid:88) {φk⊗IB} (cid:16) (cid:17){ψk⊗IB} 1.5
(cid:62) 2L (φ ,ψ )+2Q ρAB . (3)
ρA k k
k 1.0
(cid:112)
whereL (φ ,ψ )= 1|Tr(ρA[φ ,ψ ])|2/ J (φ )J (ψ ).
ρA k k 4 k k ρA k ρA k
Proof. Bydefinitionwehave
0.5
(cid:16) (cid:17) (cid:16) (cid:17)
UN ρAB +UN ρAB
(cid:88) (cid:16) {φk⊗IB} (cid:17) (cid:88) (cid:16){ψk⊗IB} (cid:17) p
(cid:62) I ρAB,φk⊗IB + I ρAB,ψk⊗IB (cid:45)1.0 (cid:45)0.5 0.5 1.0
k k
(cid:88) (cid:16) (cid:17) (cid:88) (cid:16) (cid:17) (cid:88) (cid:16) (cid:17) Figure1 Axeyistheuncertaintyanditslowerbound. Theresultin[4]is
= I ρA,φ + I ρA,ψ + [I ρAB,φ ⊗IB
k k k representedbybluethickline,thecorrespondinglowerboundisrepresented
k k k bybluedottedline;theresultin[7]isrepresentedbygreendotdashedline,
(cid:16) (cid:17) (cid:88) (cid:16) (cid:17) (cid:16) (cid:17)
−I ρA,φ ]+ [I ρAB,ψ ⊗IB −I ρA,ψ ] thelowerboundis0;thelefthandsideofourtheoremisrepresentedbyblue
k k k
dashedlineandthelowerboundofourtheoremisrepresentedbyredline.
k
(cid:62)(cid:88)2(cid:113)I(cid:0)ρA,φk(cid:1)I(cid:0)ρA,ψk(cid:1)+(cid:88)[I(cid:16)ρAB,φk⊗IB(cid:17) Insummary,wehaveestablishedanewuncertaintyrelation
k k basedontheskewinformation. Westudiedthecaseofuncer-
(cid:16) (cid:17) (cid:88) (cid:16) (cid:17) (cid:16) (cid:17)
−I ρA,φ ]+ [I ρAB,ψ ⊗IB −I ρA,ψ ] taintyrelationwiththeexistenceofaquantummemoryforthe
k k k
k bipartitequantumsystem. Ourresultshowsthatquantumcor-
(cid:88) (cid:16) (cid:17)
(cid:62) 2L (φ ,ψ )+2Q ρAB . (4) relationscanbeusedtoobtainatighterboundofuncertainty.
ρA k k
Zhihao Ma and Zhihua Chen thank Davide Girolami for useful discus-
k
sions.ThisworkissupportedbyNSFCundernumbers11275131,11371247,
ThefirstinequalityholdssinceI(ρ,R)(cid:54) J (R)(see[12]).The
ρ 11571313and11675113.
second inequality is due to the Cauchy-Schwarz inequality.
The final inequality holds because the optimal measurement
(cid:16) (cid:17)
forQ ρAB maynotbeφk orψk. 1 W.Heisenberg,Z.Phys,43,172(1927).
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6 E.P.Wigner,M.M.Yanase,Proc. Natl. Acad. Sci. U.S.A.,49,910
memoryB. Interestingly,thelowerboundcontainstwoterms,
(cid:16) (cid:17) (1963).
onetermisthequantumcorrelationQ ρAB ,theothertermis
7 S.Luo,Phys.Rev.A,72,042110(2005).
(cid:80)
kLρA(φk,ψk)whichcharacterizesthedegreeofcomplemen- 8 A.Uhlmann,ScienceChinaPhysics,MechanicsandAstronomy,59(3),
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2 c
theentropicuncertaintyrelation[3]. Thereforeourresultcan 9 K.Modi,A.Brodutch,H.Cable,etal,Rev.Mod.Phys.,84,1655-1707
beviewedasananalogueofthebipartiteentropicuncertainty (2012).
10 E.H.Lieb,Adv.Math11,267(1973).
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ρ = 2−pI + 2p−1V, where p ∈ [−1,1] and V = (cid:80)|kl(cid:105)(cid:104)kl|. 12 S.Furuichi,Phys.Rev.A,82,034101(2010).
6 4 6 13 B.Chen,S.M.Fei,ScientificReports5,14238(2015).
kl